“Never carry out any arithmetic”: the importance of structure in developing algebraic thinking

The use of tasks which ask learners to find general rules for growing figural patterns is widely reported in the algebra research literature. Such tasks are sometimes seen as a way to develop early algebra thinking. This paper looks at examples of such activities from the literature and presents a theoretical argument against the common practice of learners creating a table of values and seeking patterns within the numbers. Instead an argument is made for learners to focus on more complex examples where learners are discouraged from counting and turning the original figures into numbers. Instead, it is suggested that learners seek structure within complex examples and express what they see without carrying out any arithmetic but instead just writing down what arithmetic they would do.